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Abstract. This paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and n risky assets whose prices are log-normal diffusions. We suppose that transactions between the assets incur a cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption. Dynamic programming leads to a variational inequality for the value function. Existence and uniqueness of a viscosity solution are proved. The variational inequality is solved by using a numerical algorithm based on policies, iterations, and multigrid methods. Numerical results are displayed for n 1 and n--2.

International audienceIn the Brownian case, the links between dynamic risk measures and BSDEs have been widely studied. In this paper, we study the case with jumps. We first study the properties of BSDEs driven by a Brownian motion and a Poisson random measure. In particular, we provide a comparison theorem, under quite weak assumptions, extending that of Royer \cite{R}. We then give some properties of dynamic risk measures induced by BSDEs with jumps. We provide a representation property of such dynamic risk measures in the convex case as well as some new results on a robust optimization problem, related to the case of model ambiguityLes liens entre les EDSR et les mesures de risque dynamiques ont été largement étudiées dans le cas Brownien. On étudie dans ce papier le cas avec sauts. On étudie tout d'abord les propriétés des EDSR dirigées par un mouvement Brownien et une mesure de Poisson aléatoire : on prouve en particulier un théoréme de comparaison sous des hypothéses assez faibles qui généralise celui de Royer 2006. On donne ensuite des propriétés pour les mesures de risques dynamiques induites par les EDSR avec sauts. On établit un théoréme de représentation duale pour de telles mesures de risques dans le cas convexe. On étudie enfin un probléme d'optimisation robuste de mesure de risques associé au cas d'ambiguité de modéle

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